Department of Mathematics

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Expansion-free electromagnetic solutions of Kerr-Schild class
Debney, G. (AIP Publishing, 1974-07)
Starting with the general Kerr_Schild form of the metric tensor,d s2=_+l_l (where l is null and _ is flat space_time), a study is made for those solutions of the Einstein_Maxwell equations in which l is geodesic, shear_free, and expansion_free. It is shown that all resulting solutions must be of Petrov type [4] or type [_] and the Maxwell field must be null. Because of the expansion_free assumption there exist flat and conformally flat gauge conditions on all metrics in this class; i.e., there exist metrics of this Kerr_Schild form which are flat (or conformally flat) but are not Lorentz_related. A method is given for obtaining meaningful solutions to the field equations with the latter gauge equivalence class removed. A simple example of a radiative field of type [4] along a line singularity exhibits how solutions in this class may be generated.
Power statistics for wave propagation in one-dimension and comparison with radiative transport theory. II
Kohler, Werner E.; Papanico.G.C. (AIP Publishing, 1974-12)
We consider the one_dimensional problem of a slab having a random index of refraction and illuminated from within by a point source. We compute the expected value and the fluctuations of both the total power and power flux. These quantities, which are functions of the slab width, source location, and observation point, are determined in the limit of weak refractive index fluctuations and large slab thickness. We compare the expected values of total intensity and flux with the predictions of radiative transport theory. We also compare the results of both theories with numerical simulations.
The conditional entropy in the microcanonical ensemble
Dietz, D.; Greenberg, William (AIP Publishing, 1975-08)
The existence of the configurational microcanonical conditional entropy in classical statistical mechanics is proved in the thermodynamic limit for a class of long_range multiparticle observables. This result generalizes a theorem of Lanford for finite range observables.
Case eigenfunction expansion for a conservative medium
Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1976-02)
By using the resolvent integration technique introduced by Larsen and Habetler, the one‐speed, isotropic scattering,neutron transport equation is treated in the infinite and semi‐infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.
Functional calculus for symmetric multigroup transport operator
Greenberg, William (AIP Publishing, 1976-02)
A rigorous treatment of the symmetric multigroup transport equation is given by developing the functional calculus for the transport operator. Von Neumann spectral theory is applied to nonorthogonal cyclic subspaces, and the isometries onto C (N) are explicitly evaluated.
Solution of multigroup transport equation in L^p spaces
Greenberg, William; Sancaktar, Selim (AIP Publishing, 1976-11)
The isotropic multigroup transport equation is solved in L p , p_1, for both half range and full range problems, using resolvent integration techniques. The connection between these techniques and a spectral decomposition of the transport operator is indicated.
Vlasov theory of plasma oscillations: linear-approximation
Arthur, Michael D.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1977)
A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function _ has first and second order zeroes. Special care is taken in the study of real zeroes of _ culminating in new results for this case. For a simple zero of _ with nonvanishing imaginary part the van Kampen-Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.
The interaction function and lattice duals
Greenberg, William (AIP Publishing, 1977-10)
An interaction function is defined for lattice models in statistical mechanics. A correlation function expansion is derived, giving a direct proof of the duality relations for correlation functions.
A multiple-scales space-time analysis of a randomly perturbed one-dimensional wave equation
Kohler, Werner E. (AIP Publishing, 1977-10)
An initial value problem for one_dimensional wave propagation is considered; the medium is assumed to be randomly perturbed as a function of both space and time. The stochastic perturbation theory of Papanicolaou and Keller [SIAM J. Appl. Math. 21, 287 (1971)] is applied directly in the space-time regime to derive transport equations for the first and second moments of the solution. These equations are solved in special cases.
Uniqueness of solutions to the linearized Boltzmann equation
Garbanati, Linda F.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1978-01)
Uniqueness theorems are proved for the linearized Boltzmann equation for both the ’’exterior’’ and ’’interior’’ problems under generalized Maxwellboundary conditions. The solution space is a weighted L p space, and agrees with the space in which solutions have previously been constructed.
Transverse plasma oscillations
Arthur, Michael D.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1979)
An operator theoretic approach is used to solve the linearized Vlasov–Maxwell equations for transverse plasma oscillations. In particular, the special cases of simple and second‐order real zeros of the plasma dispersion function are treated and formulae for the amplitude of the plasma waves are presented. An existence and uniqueness theorem for the solution to the Vlasov–Maxwell transverse mode plasma equation is proved in an appendix. In a second appendix, a general characterization for the zeros of the plasma distribution function is presented for the case of any double humped equilibrium distribution.
Resolvent integration techniques for generalized transport equations
Bowden, Robert L.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1979-06)
A generalized class of ’’transport type’’ equations is studied, including most of the known exactly solvable models; in particular, the transport operator K is a scalar type spectral operator. A spectral resolution for K is obtained by contour integration techniques applied to bounded functions of K. Explicit formulas are developed for the solutions of full and half range problems. The theory is applied to anisotropicneutron transport, yielding results which are proved to be equivalent to those of Mika.
k-space function spaces
McCoy, Robert A. (Hindawi, 1980-01-01)
A study is made of the properties on X which characterize when Cπ(X) is a k-space, where Cπ(X) is the space of real-valued continuous functions on X having the topology of pointwise convergence. Other properties related to the k-space property are also considered.
Peano compactifications and property metric spaces
Dickman, R. F. (Hindawi, 1980-01-01)
Let (X,d) denote a locally connected, connected separable metric space. We say the X is S-metrizable provided there is a topologically equivalent metric ρ on X such that (X,ρ) has Property S, i.e. for any ϵ>0, X is the union of finitely many connected sets of ρ-diameter less than ϵ. It is well-known that S-metrizable spaces are locally connected and that if ρ is a Property S metric for X, then the usual metric completion (X˜,ρ˜) of (X,ρ) is a compact, locally connected, connected metric space, i.e. (X˜,ρ˜) is a Peano compactification of (X,ρ). There are easily constructed examples of locally connected connected metric spaces which fail to be S-metrizable, however the author does not know of a non-S-metrizable space (X,d) which has a Peano compactification. In this paper we conjecture that: If (P,ρ) a Peano compactification of (X,ρ|X), X must be S-metrizable. Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.
Pulse propagation in a randomly perturbed ocean: Single pulse statistics
Kohler, Werner E. (Acoustical Society of America, 1980-10)
A statistical theory of broadband single pulse propagation in a random ocean is presented. The mutual coherence function of the received signal is derived using an analysis based upon coupled mode theory. As propagation range increases, the combined effects of modal dispersion and random fluctuations spread the pulse and decompose it into a series of multiple arrivals.
Note on a role for entire functions of the classes and
Prather, Carl L. (Hindawi, 1981-01-01)
We use the B and B* operators of Levin on the Classes P and P* and a comparison principle to prove a Gauss-Lucas Theorem for differential operators. The connection with the determination of final sets for differential operators is then clarified.
Complete function spaces
McCoy, Robert A. (Hindawi, 1983-01-01)
A study is made of certain completeness properties of the space of allcontinuous real-valued functions on a space, where this function space has the compact-open topology.
Analytical solutions of model equations for two phase gas mixtures: Transverse velocity perturbations
Cavalier, J. F.; Greenberg, William (AIP Publishing, 1984)
Model equations for a dilute binary gas system are derived, using a linear BGK scheme. Complete analytical solutions for the stationary half_space problem are obtained for transverse velocity perturbations. The method of solution relies on the resolvent integration technique.
Fine topology on function spaces
McCoy, Robert A. (Hindawi, 1986-01-01)
This paper studies the topological properties of two kinds of fine topologies on the space C(X,Y) of all continuous functions from X into Y.
The effect of surface tension on the shape of a Hele-Shaw cell bubble
Tanveer, S. (AIP Publishing, 1986-11)
Numerical and asymptotic solutions are found for the steady motion of a symmetrical bubble through a parallel‐sided channel in a Hele–Shaw cell containing a viscous liquid. The degeneracy of the Taylor–Saffman zero surface‐tension solution is shown to be removed by the effect of surface tension. An apparent contradiction between numerics and perturbation arises here as it does for the finger. This contradiction is resolved analytically for small bubbles and is shown to be the result of exponentially small terms. Numerical results suggest that this is true for bubbles of arbitrary size. The limit of infinite surface tension is also analyzed.

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